A novel cepstrum-based and evolutionary optimization hybrid framework for structural parameter identification

Abstract

The accuracy and practicality of structural parameter identification remain critical challenges in Structural Health Monitoring (SHM), as many existing modal updating techniques are limited by noise sensitivity, reliance on expert judgment, and complex system identification procedures. This study introduces a novel model updating method that redefines structural parameter identification process by utilizing Cepstral Coefficients (CCs) of structural responses, which provide a compact, robust, and noise-tolerant representation of dynamic characteristics. This study pioneers the use of response-cepstrum representations in structural model updating. Unlike conventional modal characteristics, the CCs are directly and efficiently extracted from time-domain responses through signal-processing techniques, avoiding the expertise-intensive and laborious modal identification process while significantly accelerating the analysis. Exploiting the unique characteristic representation of the CCs, a customized evolutionary optimization framework is developed based on an advanced variant of the Differential Evolution (DE) algorithm for parameter identification. This development is supported by a systematic investigation of multiple effective evolutionary optimization strategies to ensure robust and accurate matching of CCs between measured and simulated responses. The proposed method was validated through both numerical simulations and experimental tests, demonstrating strong performance in identification accuracy, computational efficiency, and robustness to measurement noise as well as parameter interdependence. This method offers an efficient and scalable alternative to modal updating and establishes a new direction for feature-driven, data-centric SHM systems.

Keywords

structural dynamics efficient parameter identification cepstrum-based model updating advanced signal processing customized evolutionary optimization

Introduction

Research background and motivations

Structural health monitoring (SHM) and damage identification are increasingly critical in civil and mechanical engineering, driven by the need to ensure safety, reliability, and service longevity, alongside advancements in data-driven sensing and signal processing techniques. Complex structures, such as bridges, buildings, offshore platforms, and aerospace or mechanical systems, often operate under variable environmental and loading conditions that can lead to gradual deterioration or sudden failure (Ayati et al., 2019; Hassani and Dackermann, 2023; Thöns, 2018). Timely identification of structural changes and early damage detection are essential for preventing catastrophic failures, reducing maintenance costs, and extending structural lifespan (Hassani and Dackermann, 2023; Kaveh, 2024).

Among various SHM strategies, model updating is one of the most well-established strategies for assessing structural integrity by identifying key parameters (e.g., stiffness, damping, and mass) through their influence on dynamic response and characteristics including natural frequencies and mode shapes (Mottershead and Friswell, 1993). A widely adopted implementation of this strategy is Finite Element Model Updating (FEMU) (Ereiz et al., 2022; Mottershead et al., 2011), which employs Finite Element Models (FEMs) to simulate the dynamic behaviors of real-world structures by iteratively adjusting model parameters to minimize discrepancies between simulated and measured responses (Altunişik et al., 2019; Sarmadi et al., 2016).

In practical applications, deterministic FEMU methods based on Maximum Likelihood Methods (MLMs) have been extensively employed. These approaches define objective functions that quantify residual-based discrepancies between measured and simulated structural behaviors (Xiong et al., 2009). Various data types have been utilized to construct these objective functions, including dynamic characteristics (Durmazgezer et al., 2019; Nozari et al., 2017), static response data (Liao et al., 2012; Tchemodanova et al., 2021), or combinations of both (Sun and Xu, 2020; Zhou et al., 2017). Among them, modal characteristics, particularly natural frequencies and mode shapes, are most commonly used due to their sensitivity to structural damage and their effectiveness in capturing changes in dynamic behavior (Ereiz et al., 2022; Kudela and Matousek, 2022). Numerous studies have shown that damage-induced changes in stiffness directly affect modal characteristics, establishing them as effective indicators for structural integrity assessment and parameter identification (Guo et al., 2018; Hou et al., 2025; Mottershead and Friswell, 1993).

Alternatively, FEMU can be approached as a probabilistic inference problem that explicitly incorporates uncertainty, a perspective increasingly recognized for its theoretical rigor in practical applications (Simoen et al., 2015). This approach models uncertainties using Probability Density Functions (PDFs), with uniform and normal distributions being common in civil and mechanical engineering (Katafygiotis et al., 1998; Marwala et al., 2016). Structural parameters are typically initialized using prior PDFs derived from available knowledge and strategically updated through Bayes’ theorem. Conjugate priors (Gelman et al., 1995; Prakash and Narasimhan, 2018) are commonly employed to simplify posterior computation. Additionally, some studies have investigated approaches to minimize the impact of ambiguous prior information (Lye et al., 2021; Monchetti et al., 2024), including the use of information-theoretic criteria and scientifically bounded parameter spaces to prevent the introduction of unnecessary bias (Keshun et al., 2024; Yan and Katafygiotis, 2015).

Despite their effectiveness, traditional FEMU methods face significant challenges, including reliance on expertise-intensive model configurations, high computational cost, and limited robustness to measurement noise (Kudela and Matousek, 2022). These limitations hinder their applicability in real-time and large-scale SHM applications. To overcome this, growing attention has been directed toward data-driven surrogate models that map structural parameters directly to dynamic responses or features derived from responses, bypassing the need for extensive FEM simulations (Samadian et al., 2024). These surrogate models, such as those based on neural networks (Tang et al., 2020), enable efficient inverse analysis and help achieve rapid, adaptive structural parameter identification. Sabamehr et al. (2018) integrated neural networks with Genetic Algorithms (GAs) to correlate natural frequencies with sectional property changes in bridges, outperforming traditional matrix-based FEMU methods. Yin and Zhu (2020) developed a Bayesian neural network optimized for FEMU of a pedestrian bridge, achieving greater accuracy and efficiency than finite-difference approaches. Zhang and Sun (2021) introduced a physics-guided neural network that integrated data-driven learning with FEM principles to improve damage detection in both simulated and experimental settings. However, many existing data-driven surrogate models lack effective integration of physics-informed principles, limiting their reliability in capturing the full complexity of structural vibrations and dynamic behaviors compared to classical FEMs.

In addition to modeling approaches, optimization algorithms play a critical role in aligning simulated structural responses and characteristics, whether obtained from finite element models or data-driven surrogate models, with measured data. Among them, Evolutionary Algorithms (EAs), particularly Differential Evolution (DE), have demonstrated strong performance in solving complex, nonlinear, and high-dimensional problems typical of structural dynamics (Alkayem et al., 2018; Vesterstrom and Thomsen, 2004). DE’s conceptual simplicity, computational robustness, and effective exploration of large solution spaces make it well-suited for SHM applications (Ghannadi et al., 2023).

Numerous studies have employed classical and improved DE variants within various model updating frameworks to identify structural parameters and damage. Guedria (2020) proposed an Accelerated DE (ADE) for damage localization in plate structures using a flexibility matrix-based objective function. Wang et al. (2021) applied DE to optimize parameters in a Singular Value Decomposition (SVD)-based damage detection method for beams, validated through simulations and experiments. Zeng et al. (2023) developed a Bayesian FEMU framework incorporating the Differential Evolution Adaptive Metropolis (DREAM) algorithm, demonstrating its accuracy and efficiency on both simulated and real-world bridge structures.

The performance of DE algorithms heavily depends on managing the balance between exploration and exploitation, guided by hyperparameters such as mutation and crossover rates. To improve performance in complicated optimization scenarios, adaptive and hybrid variants have been developed, including Lévy-Flight-based Success-History Adaptive DE (LSHADE) (Tanabe and Fukunaga, 2014), and the LSHADE with Semi-Parameter Adaptation hybridized with Covariance Matrix Adaptation Evolution Strategy (CMA-ES), abbreviated as LSHADE-SPACMA (Mohamed et al., 2017). Among them, the LSHADE-SPACMA has demonstrated superior performance, particularly in solving high-dimensional, noisy, and multimodal optimization problems (Hadi et al., 2020; Mohamed et al., 2017). Although these algorithms have shown appreciable effectiveness in many problems, practical FEMU applications still require rigorous customization and constraint handling to maintain real-world optimization performance and accommodate complex structural parameter spaces (Alkayem et al., 2018; Ghannadi et al., 2023).

Another critical consideration, as introduced above, is that a large portion of FEMU studies in the literature rely on modal analysis of structural characteristics such as natural frequencies and mode shapes, which serve as the primary optimization objectives. However, these approaches are heavily dependent on advanced structural identification algorithms, such as Frequency Domain Decomposition (FDD) (Brincker and Zhang, 2009) and Stochastic Subspace Identification (SSI) (Peeters and De Roeck, 1999). These techniques often require extensive hyperparameter tuning and expert-driven calibration, which are typically carried out offline and involve manual, labor-intensive procedures with limited automation. This results in a “two-step” parameter identification process that can hinder real-time or practical deployment (Ereiz et al., 2022; Lye et al., 2021). Moreover, their performance is often highly sensitive to measurement noise and external excitations. Under noisy environmental conditions, these limitations can significantly reduce the reliability and accuracy of the identified structural parameters (Rainieri and Fabbrocino, 2014).

This study is motivated by the aforementioned limitations and the underexplored challenges associated with modal-analysis-based FEMU frameworks and evolutionary optimization algorithms, aiming to develop a practically efficient and robust method for structural parameter identification.

Present study and main contributions

This study presents a novel FEMU framework that utilizes the mathematical properties and unique modal representation capabilities of Cepstral Coefficients (CCs), extracted from structural acceleration responses, to enable efficient and robust structural parameter identification. A new feature-based objective function is formulated as a physically grounded, location-weighted discrepancy between measured and simulated CCs, effectively capturing variations in dynamic behavior. To solve the resulting optimization problem, the LSHADE-SPACMA algorithm is adopted and tailored with domain-specific constraints and initialization strategies, enhancing accuracy, convergence speed, and robustness.

While CCs have shown promise as compact and damage-sensitive features in structural diagnostics (Li et al., 2025; Mei et al., 2019; Morgantini et al., 2021), their integration into structural parameter identification has remained largely unexplored. This study fills that gap by developing a theory-driven, noise-resilient, and computationally efficient CC-based FEMU method, offering strong potential for real-world SHM applications.

The primary contributions of this research are as follows:

(1) The CCs of structural responses, grounded in vibration theory and sensitive to damage, exhibit a linear summation form that isolates global modal characteristics, such as natural frequencies, from disturbing local variations caused by noise and excitation variability. By utilizing these mathematical and physical properties, we formulate a new location-weighted discrepancy objective function based on the simulated and measured CCs, enhancing robustness to noise and excitation variability. This formulation supports a highly noise-resilient and computationally efficient structural parameter identification process.

(2) Integrated with digital signal processing, the CCs are automatically and efficiently extracted from structural vibration responses, enabling seamless configuration into the FEMU framework. This automation can help avoid the labor-intensive system identification procedures with a fully practical and efficient “one-step” process. This method enhances the intelligence and adaptability of the model updating for practical engineering applications by streamlining the workflow and improving robustness to common system identification challenges, including hyperparameter instability and environmental disturbances.

(3) The LSHADE-SPACMA algorithm is enhanced through the integration of a baseline-condition reference strategy that scientifically quantifying parameter changes, along with the implementation of appropriate prior ranges and constraint settings during parameter identification. These enhancements are adapted to real-world engineering contexts where mass, damping, and stiffness properties are often partially known, vary in reliability, and differ in significance. These are particularly effective in situations where the baseline condition of the structure is uncertain or inadequately defined, facilitating more accurate detection of parameter changes indicative of potential damage in subsequent assessments.

(4) The performance of the proposed CC-based FEMU method was systematically validated by replacing the integrated LSHADE-SPACMA algorithm with several classical optimization algorithms, including GA, Particle Swarm Optimization (PSO), and established DE variants, for benchmarking and comparison. Both numerical simulations and experimental case studies confirm the method’s superior accuracy, robustness, and practical viability for structural parameter identification, including its effectiveness in mitigating the common FEMU challenge of stiffness-damping coupling interference.

Methodology

This section presents the methodological framework of the proposed FEMU method, structured to systematically introduce its theoretical foundation and practical implementation potential: The first part establishes the theoretical rationale for employing the CCs extracted from structural acceleration responses, demonstrating their physical interpretability and relevance for model updating. Next, the LSHADE-SPACMA algorithm is introduced, outlining its adaptive mechanisms and its suitability for high-dimensional, noise-prone structural identification tasks. The following part details the customized implementation strategies developed to enhance method robustness and generalizability, including the reference-based baseline condition identification and the application of physically informed parameter constraints. Finally, the computational efficiency and deployment advantages of the proposed method are discussed, highlighting its strong feasibility for real-world SHM applications.

The cepstral coefficients of structural acceleration response

Let us consider the equations of motion of an $N_{d}$ degree-of-freedom (DOF) model of a linear time-invariant system:

M \ddot{y} (t) + C \dot{y} (t) + K y (t) = u (t)

(1)

where

M \in R^{N_{d} \times N_{d}}

C \in R^{N_{d} \times N_{d}}

and

K \in R^{N_{d} \times N_{d}}

are namely the mass, damping and stiffness matrices. The vectors

y (t) \in R^{N_{d}}

\dot{y} (t) \in R^{N_{d}}

, and

\ddot{y} (t) \in R^{N_{d}}

represent nodal displacement, velocity and acceleration, respectively, while

u (t) \in R^{N_{d}}

represents the input vector, containing

N_{d}

nodal external excitations at time

t

. Performing the z-transform to the acceleration vector

\ddot{y} (t)

, we can obtain the z-domain acceleration

A_{d} (z)

for the

d

th DOF (

d = 1

,…,

N_{d}

) as follows:

A_{d} (z) = \sum_{j = 1}^{N_{d}} H_{I} {(z)}_{d, j} U_{d} (z)

(2)

where

H_{I} {(z)}_{d, j}

represents the

(d, j)

th element of the structural dynamics inertance matrix in the z-domain, while

U_{d} (z)

denotes the z-transform of the excitation input

u_{d} (t)

applied at the

d

th DOF. The element

H_{I} {(z)}_{d, j}

is given by:

H_{I} {(z)}_{d, j} = \sum_{l = 1}^{N_{d}} \frac{ϕ_{d, l} ϕ_{j, l} (1 - z^{- 1}) (1 - P_{l} z^{- 1})}{(1 - e^{λ_{l} Δ t} z^{- 1}) (1 - e^{λ_{l}^{*} Δ t} z^{- 1})}

(3)

where

λ_{l}

and

λ_{l}^{*}

are the complex conjugate eigenvalues linked to the

l^{t h}

mode shape of the system, while

ϕ_{d, l}

and

ϕ_{j, l}

are the elements corresponding to the

d^{t h}

and

j^{t h}

locations of the

l^{t h}

mode shape, respectively. The term

Δ t

represents the sampling interval of the recorded acceleration time histories. The term

P_{l}

is a function of modal characteristics, given by:

P_{l} = \frac{e^{- ξ_{l} ω_{l} T} \cos (ω_{d a m p, l} Δ t - ξ_{l})}{\sqrt{1 - ξ_{l}^{2}}}

(4)

where

ω_{d a m p, l} = ω_{l} \sqrt{1 - ξ_{l}^{2}}

is the damped natural frequency. By mathematically reformulating the right-hand side of equation (3) into a root product form,

A_{d} (z)

can be expressed as:

A_{d} (z) = \frac{(1 - z^{- 1}) \prod_{l = 1}^{R} (1 - Z_{l}^{(d)} z^{- 1})}{\prod_{l = 1}^{N_{d}} (1 - e^{λ_{l} Δ t} z^{- 1}) (1 - e^{λ_{l}^{*} Δ t} z^{- 1})}

(5)

where the terms

Z_{l}^{(d)}

for

l = 1, \dots, R

are roots of the following equation:

\sum_{j = 1}^{N_{d}} U_{j} (z) \sum_{l = 1}^{N_{d}} ϕ_{d, l} ϕ_{j, l} (1 - P_{a, l} z^{- 1}) \prod_{\begin{array}{l} k = 1 \\ k \neq l \end{array}}^{N_{d}} (1 - e^{λ_{k} Δ t} z^{- 1}) (1 - e^{λ_{k}^{*} Δ t} z^{- 1}) = 0

(6)

Thus, it can be observed that $Z_{l}^{(d)}$ involves key structural elements, including the global characteristics of natural frequencies $ω_{l}$ for $l = 1, \dots, N_{d}$ , as well as the local mode-shape element $ϕ_{d, l}$ linked to the location $d$ .

As derived in Li et al. (2024) and Morgantini et al. (2021), the result of inverse z-transform of the logarithm of the acceleration power spectrum of location $d$ is essentially the corresponding CCs in the quefrency domain, expressed as:

Z^{- 1} {\ln ({| A_{d} (z) |}^{2})} = c_{d} [q] = θ [q] + γ_{d} [q]

(7)

where

c_{d} [q]

represents the

q

th CC extracted from the response at the

d

th DOF, and

q

denotes the quefrency index (

q = 1, 2, \dots

). The two terms

θ [q]

and

γ_{d} [q]

are given by:

{\begin{cases} θ [q] = \frac{1}{q} \sum_{l = 1}^{N_{d}} 2 e^{- ξ_{l} ω_{l} Δ t q} \cos (ω_{d a m p, l} Δ t q) - 1 \\ γ_{d} [q] = - \frac{1}{q} \sum_{l = 1}^{R} Z_{l}^{(d) q} \end{cases}

(8)

Only CCs for $q > 0$ are considered, as $c_{d} [q]$ at $q = 0$ is dominated by excitation inputs and sensor settings, while $c_{d} [q]$ for $q < 0$ are zeros (Li et al., 2024). The magnitude of $c_{d} [q]$ decays exponentially with increasing $q$ , gradually reducing its informational contribution. Therefore, in this study, the parameter identification process is conducted using the first 100 CCs, i.e., $c_{d} [1]$ , $c_{d} [2]$ ,…, $c_{d} [100]$ , since the later terms generally offer negligible useful information to the identification process due to their exponential decay property.

It can be found that for the acceleration responses recorded under identical conditions (e.g., the undamaged state) at the location $d$ , variations in $c_{d} [q]$ primarily stem from $γ_{d} [q]$ , specifically influenced by the combination of the $R$ roots $Z_{l}^{(d)}$ ( $l = 1, \dots, R$ ). Consequently, $γ_{d} [q]$ contains local structural characteristic, particularly the local mode shape element $ϕ_{d, l}$ , which varies with the location $d$ . In contrast, $θ [q]$ is associated solely with global structural characteristics, such as natural frequencies and damping ratios, and thus remains consistent across different locations.

Accordingly, a FEMU framework designed to minimize the error between the CCs of simulated and measured acceleration responses is inherently consistent with minimizing discrepancies in key structural characteristics, including natural frequencies, damping ratios, and mode shapes.

The LSHADE-SPACMA-based optimization framework

The objective function of the CC-based updating framework

As noted, the CC $c_{d} [q]$ s extracted from the acceleration response recorded at location $d$ , with its variation primarily influenced by $γ_{d} [q]$ . Variations in $γ_{d} [q]$ arise from a combination of influences, including localized excitation conditions, measurement noise, and environmental disturbances. These sources introduce stochastic variability into $c_{d} [q]$ , which can hinder the updating framework’s ability to insightfully analyze key structural characteristics such as natural frequencies and mode shapes.

To mitigate these effects, the proposed CC-based FEMU framework introduces an objective function that computes weighted location-specific discrepancies between simulated and measured CCs. This formulation improves robustness by emphasizing consistent structural patterns while reducing the impact of localized noise and excitation variability. This objective function is expressed as:

J (θ) = \frac{1}{N_{d} \cdot Q} \sum_{d = 1}^{N_{d}} \sum_{q = 1}^{Q} w_{d} {(c_{d}^{(m)} [q] - c_{d}^{(s)} [q; θ])}^{2}

(9)

where

θ

represent the structural parameters such as stiffness, damping ratio and mass that to be identified;

c_{d}^{(m)} [q]

represents the CCs extracted from the measured acceleration response, while

c_{d}^{(s)} [q; θ]

denote the simulated CCs that depend on

θ

. The term

Q

represents the number of CCs considered for the identification. The term

w_{d}

represents the weight links to the location

d

, which is determined by the level of variance attributed by the measured CC

c_{d}^{(m)} [q]

, as follows:

w_{d} = \frac{C_{0}}{\frac{1}{Q - 1} \sum_{q = 1}^{Q} {(c_{d}^{(m)} [q] - {\bar{c}}_{d, Q}^{(m)})}^{2}}

(10)

where

{\bar{c}}_{d, Q}^{(m)}

represent the average of the measured sequential CCs

c_{d}^{(m)} [q]

for

q = 1, \dots, Q

at the location

d

. The constant

C_{0}

is determined by the following condition on the weight

w_{d}

\sum_{d = 1}^{N_{d}} w_{d} = 1

(11)

Accordingly, the weight $w_{d}$ is defined as inversely proportional to the variance of the CCs at location $d$ . As introduced in Eqs. (7) and (8), the variance (or variation) of the local CCs is strongly influenced by local excitation and noise, as well as the local mode shape element $ϕ_{d, l}$ in $γ_{d} [q]$ . This weighting scheme therefore assigns greater importance to CCs with lower local variance, effectively reducing the influence of high-variance components typically associated with noise and localized excitations. Importantly, although the contribution of the local mode shape element $ϕ_{d, l}$ associated with noisy or highly excited regions may be slightly attenuated, the global mode shape information is preserved, as the weights $w_{d}$ are normalized to sum to one, as shown in equation (11). Through this weighted objective function, the impact of stochastic variations and noise in the CCs can be minimized, thereby enhancing the reliability of structural parameter identification and model updating.

The LSHADE-SPACMA algorithm

The core optimization engine of the proposed CC-based FEMU framework is the LSHADE-SPACMA algorithm, considered for its demonstrated computational efficiency and robustness in solving complex structural parameter identification problems. Its essential DE mechanism is characterized by strong exploration capability and a simple yet effective mutation strategy, enabling efficient navigation of complex search spaces and reducing the risk of premature convergence, a particularly important feature for SHM tasks (Ghannadi et al., 2023). The LSHADE component enhances performance by employing adaptive control of scaling factors through stochastic and feedback-driven adjustments, significantly improving convergence and stability across diverse problem landscapes (Tanabe and Fukunaga, 2014). Moreover, the LSHADE-SPACMA strategically combines a semi-parameter adaptation mechanism of LSHADE-SPA (Mohamed et al., 2017; Tanabe and Fukunaga, 2014) with a modified CMA-ES configuration (Hansen, 2006), further strengthening the algorithm’s exploration capabilities via an efficient crossover operation, promoting diversity and avoiding premature convergence in complex, high-dimensional search spaces.

This synergistic combination of the two mechanisms makes the LSHADE-SPACMA particularly suitable for structural identification tasks where the search landscape is non-convex, ill-conditioned, or contains multiple optima. For a comprehensive description of the algorithm’s theoretical foundations and implementation details, readers are referred to Mohamed et al. (2017). Comparative studies with detailed performance evaluations (provided in Results) validate that the LSHADE-SPACMA consistently outperforms well-established algorithms including the GA and PSO, as well as its predecessors of the standard CMA-ES and LSHADE, in terms of global search capability, convergence speed, and solution robustness.

Customized implementation strategies of the FEMU method

While DE-based algorithms have demonstrated strong potential in various SHM tasks including parameter identification, sensor placement, and damage localization, most existing studies are limited to specific structural configurations or simplified scenarios, often focusing solely on stiffness parameter identification. Moreover, robust and generalizable implementation strategies are essential for reliably capturing damage-related parameter changes in real-world applications, particularly under uncertain prior information and environmental variability. To address these practical challenges, this study introduces a set of customized strategies for deploying the LSHADE-SPACMA algorithm in conjunction with the CC-based objective function, as detailed below:

Identification approaches for baseline and damaged conditions

To systematically detect structural parameter changes indicative of damage, this study adopts a reference-based strategy using two complementary implementation schemes (termed as Approaches 1 and 2): Approach 1 performs direct estimation of structural parameters via the proposed CC-based FEMU method for both baseline and damaged states; Approach 2 estimates relative percentage changes in parameters with respect to the previously identified baseline, enabling focused assessment of damage-induced variations.

For baseline identification, parameters are initialized using prior structural knowledge; for damaged conditions, the baseline identification results are used as initial values to maintain continuity and to reduce search space ambiguity.

The accuracy and stability of both approaches, particularly in detecting localized stiffness degradation, are evaluated through Case Study 1. This dual-strategy framework allows for a comprehensive assessment of the FEMU method’s optimization performance under differing value search spaces (absolute values vs relative changes).

Prioritization of stiffness and damping parameters

In the LSHADE-SPACMA optimization framework, stiffness is prioritized for optimization due to its dominant impact on structural damage, followed by damping, which is critical for capturing the dynamic response under operational conditions. Mass is typically treated as a known parameter in many FEMU problems due to its minimal influence compared to stiffness and damping, and in practical applications, it is often considered fixed or reference-based as it tends to remain stable over time (Ereiz et al., 2022; Kudela and Matousek, 2022). This strategy is considered in Case Study 1 that can minimizes computational burden, allowing the optimization process to focus on the more critical parameters (stiffness and damping), thereby accelerating convergence and improving efficiency.

Proper constraints in parameter identification process

To ensure physical realism and algorithmic stability, constraints are applied to stiffness and damping parameters, which are most affected by structural damage, while mass typically experiences minor changes due to environmental disturbances, as discussed in Case Study 2.

During baseline model updating, stiffness and damping are estimated without constraints to capture the undamaged state accurately. For damage scenarios, appropriate bounded constraints are imposed to ensure physically plausible structural responses, as detailed below:

Stiffness is allowed to decrease by up to 80% (to 20% of the baseline value), and increase by no more than 20% to account for modeling uncertainty while preventing nonphysical results.

Damping ratios are permitted to increase by up to 200% or decrease by up to 50%, based on evidence that damage, particularly in shear-type structures, typically leads to increased damping. For example, damping ratios in undamaged concrete and steel frames typically range between 1 – 5%, but can exceed 10% with maximum being up to 200% due to shear cracking or joint failure (American Society of Civil Engineers, 2023; Brownjohn, 2007; Yuen and Kuok, 2010).

These implementation strategies utilize the superior robustness and efficiency of the LSHADE-SPACMA algorithm compared to other DE-based and evolutionary algorithms. Detailed implementation procedures and comparative results with alternative optimization algorithms are presented in Results. The overall workflow of the CC-based FEMU method is illustrated in the flowchart shown in Figure 1.

Figure 1.

Implementation flowchart of the CC-based FEMU method.

Computational efficiency of the CC-based FEMU method

The proposed CC-based FEMU method achieves high computational efficiency by directly extracting CCs from structural acceleration responses, eliminating the need for intermediate modal identification that can be time-intensive and sensitive to noise. This enables seamless integration with the LSHADE-SPACMA algorithm and simplifies the workflow, while preserving sensitivity to structural changes, thereby yielding an efficient and robust optimization framework for structural parameter identification.

The method was implemented in MATLAB and tested on a standard desktop computer (Intel® Core™ 3.89 GHz CPU, 16 GB RAM). In the 8-DOF shear-type system (Case study 1), each identification run, comprising simulation, CC extraction, objective evaluation, and parameter updating, completes in approximately 270–290 s. In the experimental 3-story frame (Case study 2), runtimes range from 200–220 s. Comparative convergence results with benchmark algorithms are presented in Results, further validating the method’s practical efficiency and suitability for single-CPU environments.

Results

8 DOF shear-type system – Case study 1

Dataset simulation and basic settings

In this case, an 8-DOF shear-type lumped mass FEM of a structural system was simulated to validate the developed CC-based FEMU method, as presented in Figure 2, including a baseline condition (undamaged) and 7 damaged scenarios. In the baseline configuration, each vertical element (DOF) is assigned by a stiffness of $k_{0, d} = k_{0} = 25, 000 N / m$ and a mass of $m_{0, d} = m_{0} = 1 kg$ , for $d = 1, \dots, 8$ . The 7 damaged scenarios are set by modifying the stiffness of specific elements to simulate various damage locations and severity levels, as detailed in Table 1. A uniform modal damping ratio of 1% is applied to all eight vibrational modes (i.e., $ξ_{0, d} = ξ_{0} = 0.01$ for $d = 1, \dots, 8$ ).

Figure 2.

The simulated 8 DOF structural system (baseline condition).

Table 1.

Simulated baseline (undamaged) and damaged scenarios of the 8 DOF system.

Scenarios	Condition	Types of anomalies
1	Undamaged	Baseline scenario
2	Damaged	$k_{d} = 0.9 k_{0}$ for $d = 1$
3	Damaged	$k_{d} = 0.9 k_{0}$ for $d = 3$
4	Damaged	$k_{d} = 0.9 k_{0}$ for $d = 5$
5	Damaged	$k_{d} = 0.9 k_{0}$ for $d = 7$
6	Damaged	$k_{d} = 0.85 k_{0}$ for $d = 7$
7	Damaged	$k_{d} = 0.9 k_{0}$ for $d = 3, 7$
8	Damaged	$k_{d} = 0.9 k_{0}$ for $d = 2, 8$

In each scenario, all 8 DOFs were independently excited using zero-mean Gaussian White Noise (GWN) inputs, applied under a zero-order-hold assumption with a Root Mean Square (RMS) of 100 N. The excitations lasted 100 seconds and were sampled at 200 Hz. To emulate realistic measurement conditions, 10% RMS Gaussian noise was added to the simulated acceleration responses. The CCs were then extracted from the noisy acceleration data following the procedure outlined in Methodology.

The primary objective of this case study is to evaluate the method’s ability to identify stiffness variations indicative of structural damage. Although damping ratios are fixed across scenarios, they are treated as unknowns to assess the method’s robustness. Mass parameters are assumed known, given their relatively minor influence compared to stiffness and damping.

As established in matrix perturbation theory of structural dynamics (Chen, 2005), changes in the location or magnitude of stiffness reductions simultaneously affect both natural frequencies and mode shapes. Moreover, relying solely on natural frequency variations for FEMU often yields ill-posed inverse problems due to insufficient sensitivity to spatial damage characteristics (Mottershead et al., 2011; Mottershead and Friswell, 1993). In this study, the developed method minimizes the discrepancy between measured and FEM-generated CCs, as formulated in equation (9), which inherently capture modal information from both the frequencies and mode shapes across all DOFs (see Methodology). Theoretically, distinct stiffness reduction patterns produce unique CC trends, enabling reliable discrimination between damage scenarios. As illustrated in Figure 3, the CCs from different DOFs in the 8-DOF system exhibit clear, scenario-specific deviations between baseline and damaged conditions, validating the sensitivity and discriminatory power of the CC-based FEMU framework.

Figure 3.

The trends of CCs of the baseline condition and different damaged scenarios, specifically for DOF 1 (a), DOF 3 (b), DOF 5 (c), and DOF 7 (d) of the 8 DOF shear-type structural system.

To comprehensively evaluate the CC-based FEMU method, two identification approaches were implemented as introduced previously: 1) direct estimation of stiffness and damping ratios (Approach 1) and 2) estimation of their percentage changes relative to known baseline values (Approach 2). Although baseline parameters are known for this simulated case, Approach 1 was first applied to recover them, with initial values uniformly sampled within ±20% of their true values. This setup simulates practical uncertainties in prior structural knowledge, and facilitates a systematic comparison of the optimization performance of various algorithms. For subsequent damage scenario identification, parameters were initialized using the identified baseline results to ensure consistency across scenarios. This allows for direct comparison of parameter changes, particularly stiffness reductions, relative to the undamaged reference.

Optimization performance comparison for different algorithms

The performance of the CC-based FEMU method, integrated with the LSHADE-SPACMA, was evaluated against four established optimization algorithms. These include the LSHADE and CMA-ES algorithms, which form the two core components of the LSHADE-SPACMA, as well as the classical GA (Tang et al., 1996) and PSO (Wang et al., 2018), both of which have been widely used in FEMU with demonstrated effectiveness.

The comparative analysis focused on convergence speed, identification accuracy, and solution stability under the baseline condition (Scenario 1). Each algorithm was applied to identify stiffness and damping ratios by minimizing the location-weighted error between the simulated and reference CCs across all 8 DOFs. Table 2 summarizes the initialization settings and identified results for each algorithm within the CC-based FEMU framework. To ensure reliability, each algorithm was executed over 50 independent runs using 50 separately generated reference CC sets, with the GWN excitation re-simulated at each iteration during the updating process. The corresponding convergence behavior is illustrated in Figure 4.

Table 2.

Identified stiffness k’_0,d and damping ratios ξ’_0,d (d =1, … ,8) of the 8 DOF system in its baseline condition. Values in brackets indicate the min and max of either the initialized parameters (first 2 rows) or the identified results using the GA, PSO, CMA-ES, LSHADE, and the LSHAD-SPACMA, obtained from 50 runs of each. The values in bold represent the mean of the identified results across these 50 runs.

		DOF 1	DOF 2	DOF 3	DOF 4	DOF 5	DOF 6	DOF 7	DOF 8
Ranges of initialized parameter values	$k_{0, d}^{'}$	[2.131e4, 2.826e4]	[2.098e4, 2.818e4]	[2.103e4, 2.873e4]	[2.126e4, 2.908e4]	[2.016e4, 2.885e4]	[2.187e4, 2.952e4]	[2.183e4, 2.847e4]	[2.141e4, 2.927e4]
Ranges of initialized parameter values	$ξ_{0, d}^{'}$	[0.0081, 0.0119]	[0.0080, 0.0119]	[0.0082, 0.0118]	[0.0081, 0.0120]	[0.0081, 0.0119]	[0.0080, 0.0118]	[0.0081, 0.0120]	[0.0080, 0.0119]
GA	$k_{0, d}^{'}$	2.578e4 [2.482e4, 2.656e4]	2.561e4 [2.477e4, 2.649e4]	2.475e4 [2.416e4, 2.541e4]	2.481e4 [2.429e4, 2.550e4]	2.538e4 [2.470e4, 2.602e4]	2.479e4 [2.413e4, 2.544e4]	2.468e4 [2.406e4, 2.529e4]	2.485e4 [2.427e4, 2.543e4]
GA	$ξ_{0, d}^{'}$	0.0106 [0.0095, 0.0114]	0.0096 [0.0089, 0.0104]	0.0093 [0.0085, 0.0104]	0.0105 [0.0096, 0.0113]	0.0103 [0.0092, 0.0113]	0.0094 [0.0086, 0.0101]	0.0107 [0.0095, 0.0115]	0.0105 [0.0094, 0.0114]
PSO	$k_{0, d}^{'}$	2.583e4 [2.488e4, 2.682e4]	2.580e4 [2.482e4, 2.671e4]	2.578e4 [2.479e4, 2.667e4]	2.443e4 [2.361e4, 2.535e4]	2.447e4 [2.368e4, 2.542e4]	2.471e4 [2.384e4, 2.566e4]	2.439e4 [2.349e4, 2.532e4]	2.535e4 [2.448e4, 2.629e4]
PSO	$ξ_{0, d}^{'}$	0.0108 [0.0096, 0.0117]	0.0105 [0.0096, 0.0113]	0.0094 [0.0083, 0.0108]	0.0097 [0.0089, 0.0106]	0.0106 [0.0094, 0.0117]	0.0093 [0.0084, 0.0105]	0.0105 [0.0093, 0.0116]	0.0106 [0.0094, 0.0115]
CMA-ES	$k_{0, d}^{'}$	2.531e4 [2.506e4, 2.568e4]	2.479e4 [2.436e4, 2.529e4]	2.556e4 [2.491e4, 2.618e4]	2.543e4 [2.494e4, 2.589e4]	2.489e4 [2.445e4, 2.532e4]	2.480e4 [2.436e4, 2.524e4]	2.491e4 [2.449e4, 2.533e4]	2.521e4 [2.492e4, 2.559e4]
CMA-ES	$ξ_{0, d}^{'}$	0.0104 [0.0098, 0.0109]	0.0095 [0.0089, 0.0102]	0.0094 [0.0087, 0.0102]	0.0105 [0.0096, 0.0113]	0.0103 [0.0095, 0.0110]	0.0095 [0.0088, 0.0102]	0.0104 [0.0097, 0.0110]	0.0103 [0.0096, 0.0109]
LSHADE	$k_{0, d}^{'}$	2.518e4 [2.493e4, 2.542e4]	2.488e4 [2.463e4, 2.513e4]	2.490e4 [2.468e4, 2.526e4]	2.517e4 [2.485e4, 2.540e4]	2.513e4 [2.488e4, 2.545e4]	2.491e4 [2.472e4, 2.512e4]	2.489e4 [2.469e4, 2.517e4]	2.511e4 [2.482e4, 2.536e4]
LSHADE	$ξ_{0, d}^{'}$	0.0096 [0.0090, 0.0103]	0.0105 [0.0097, 0.0110]	0.0095 [0.0088, 0.0104]	0.0098 [0.0092, 0.0106]	0.0103 [0.0096, 0.0107]	0.0098 [0.0091, 0.0106]	0.0099 [0.0092, 0.0108]	0.0105 [0.0097, 0.0112]
LSHAD-SPACMA	$k_{0, d}^{'}$	2.515e4 [2.495e4, 2.539e4]	2.494e4 [2.489e4, 2.509e4]	2.516e4 [2.494e4, 2.531e4]	2.491e4 [2.477e4, 2.510e4]	2.495e4 [2.481e4, 2.508e4]	2.506e4 [2.488e4, 2.522e4]	2.509e4 [2.495e4, 2.524e4]	2.497e4 [2.486e4, 2.511e4]
LSHAD-SPACMA	$ξ_{0, d}^{'}$	0.0103 [0.0095, 0.0110]	0.0098 [0.0093, 0.0105]	0.0096 [0.0090, 0.0104]	0.0103 [0.0097, 0.0109]	0.0097 [0.0091, 0.0106]	0.0104 [0.0096, 0.0111]	0.0097 [0.0091, 0.0104]	0.0098 [0.0091, 0.0106]

Figure 4.

Convergence performance among the 5 considered optimization algorithms.

As shown in Table 2, the LSHADE-SPACMA and LSHADE achieved the highest identification accuracy and stability for both baseline stiffness and damping ratios, while the CMA-ES and GA delivered moderately accurate results, with the PSO exhibiting the lowest accuracy. For convergence speed (Figure 4), the LSHADE-SPACMA and CMA-ES achieve relatively faster convergence than the other algorithms. Although the CMA-ES converges quickly, it tends to terminate early with a higher objective loss, indicating limited further improvement.

To strengthen the analysis, Mann–Whitney U-tests were conducted on the identified stiffness parameters (acknowledging their non-strict normal distribution) between the LSHADE-SPACMA and the 4 benchmark algorithms. Based on the 50 independent runs averaged over 8 DOFs, the p-values against the GA, PSO, CMA-ES, and LSHADE are 0.022, 0.013, 0.089, and 0.107, respectively. At the 0.05 significance level, these results confirm statistically significant improvements over the GA and PSO, while the comparisons with the CMA-ES and LSHADE indicate moderate yet consistent performance gains, in line with their relatively accurate identification results reported in Table 2.

These performance differences can be attributed to the following several key factors:

Firstly, the LSHADE-SPACMA outperforms the CMA-ES in accuracy probably due to its integration of LSHADE’s strong global exploration, effective in navigating the stiffness space across all 8 DOFs, with CMA-ES’s efficient local exploitation through adaptive covariance. In contrast, the CMA-ES alone lacks sufficient global search capability, particularly when handling stiffness and damping parameters that have coupled effects (see Case study 1).

Secondly, the LSHADE-SPACMA’s superior convergence over the LSHADE probably stem from CMA-ES’s adaptive covariance matrix, which improves search efficiency in high-dimensional spaces. Although the LSHADE alone still attains appreciable accuracy given sufficient iterations, it requires more time and exhibits slightly greater variability, as seen in its wider max-min range across the 50 independent runs.

Finally, the lower accuracy of PSO and slower convergence of GA can be attributed to their limited adaptability. The PSO tends to prematurely converge due to rapid swarm contraction and lacks fine-grained local search capabilities, while GA’s fixed genetic operators are less effective in refining solutions in high-dimensional, nonlinear spaces. As a result, both struggle with parameter interdependencies and yield higher error variability in this 16-parameter identification task.

Parameter identification of structural damage conditions

Following the comparative evaluation of optimization algorithms, this section examines the effectiveness of the CC-based FEMU framework in identifying stiffness and damping parameters under damaged conditions (Scenarios 2 to 8), with emphasis on stiffness degradation. The two identification approaches were both applied to each case: Approach 1 (direct parameter estimation) and Approach 2 (estimation of percentage changes relative to the baseline).

For both approaches, parameter initialization in damaged scenarios was based on the mean values identified for the baseline (Table 2, last row). In Approach 2, percentage changes were initialized to one, representing no initial deviation from the baseline. Constraints on parameter variation were applied consistently for both approaches to ensure comparable solution spaces.

Bar charts in Figures 5 and 6 summarize the identified stiffness and damping ratios across all damaged scenarios using Approaches 1 and 2, respectively. Each case was evaluated over 50 independent runs to ensure statistical robustness, with error bars indicating the maximum and minimum identified values. The results reveal the following key insights:

(1) The CC-based FEMU method accurately identifies all seven damage scenarios involving stiffness reduction, regardless of whether direct parameter estimation (Approach 1) or percentage-change identification (Approach 2) is used. Notably, in Scenarios 6 to 8, which involve higher damage severity or multiple locations, Approach 2 demonstrates slightly better accuracy and stability. This improvement likely results from its normalized and constrained search space, which enhances optimization focus compared to the broader search domain in Approach 1.

(2) As damage severity increases (Scenarios 6–8), the identified damping ratios consistently rise relative to the baseline, regardless of the identification approach, despite no intentional changes in damping. This trend coincides with slight underestimations of stiffness reductions, e.g., ∼14% versus 15% in Scenario 6 and ∼9% versus 10% in Scenarios 7 and 8. These discrepancies probably result from stiffness-damping coupling effects in structural dynamics (Mottershead and Friswell, 1993), where elevated damping estimates can partially offset the dynamic changes caused by damage. Since increased damping lowers damped natural frequencies, effects similar to stiffness loss, the optimizer may converge to higher damping values to match the observed CC trends (Eq. (8)). Although actual damping remains unchanged, this compensatory behavior is common in data-driven identification frameworks using evolutionary algorithms or neural networks (Marwala, 2010), and should be interpreted as a modeling artifact rather than a true physical change.

(3) Despite the stiffness-damping coupling effects, the proposed method accurately localizes and quantifies stiffness reductions at the affected DOFs. The slight underestimation observed reflects an inherent trade-off in simultaneous parameter optimization rather than a methodological limitation. Importantly, the consistent identification performance highlights the informativeness and sensitivity of CCs, which inherently capture both global (e.g., natural frequencies) and local (e.g., mode shapes) structural characteristics. These properties validate the suitability of CCs as a robust and discriminative objective for structural parameter identification within the FEMU framework.

Figure 5.

Identified stiffness (a) and damping ratio (b) parameters using Approach 1 across the 7 damaged scenarios of the 8-DOF system. Blue bars denote the mean values from 50 runs; error bars represent the observed min and max extrema. Red dashed lines mark the baseline values (k₀ = 25,000 N/m, ξ₀ = 0.01); red arrows indicate damaged DOFs with annotated relative ratios to both the baseline stiffness k₀ and the identified undamaged reference k’_0,d (d = 1,…,8).

Figure 6.

Identified parameter changes using Approach 2 for stiffness (a) and damping ratios (b) across the 7 damaged scenarios, relative to the identified baseline values of the 8-DOF system. Blue bars show mean results over 50 runs; error bars indicate the observed min and max extrema. Red dashed lines denote zero change relative to the identified baseline, i.e., 100% × k’_0,d and 100% × ξ’_0,d (d = 1,…,8), respectively. Red arrows mark damaged DOFs with annotated percentage changes relative to the identified baseline k’_0,d (d = 1,…,8) and true baseline k₀.

Experimental 3-story frame structure – Case study 2

Experiment setup and dataset properties

To further evaluate the robustness and applicability of the proposed FEMU method, experimental acceleration data from an experimental three-story frame tested at Los Alamos National Laboratory (LANL), New Mexico, USA, were analyzed, shown in Figure 7(a). The dataset is publicly available for download at https://institute.lanl.gov/ei/software-and-data/data. This three-story frame consists of aluminum floor plates and vertical columns joined by bolted connections, with each of the three plates (30.5 × 30.5 × 2.5 cm) connected by four columns (17.7 × 2.5 × 0.6 cm). A suspended column (15.0 × 2.5 × 2.5 cm) hangs from the top floor, contacting a floor-mounted bumper and introducing potential nonlinear or impact dynamics. A lateral band-limited random excitation in the 20–150 Hz range was applied through an electrodynamic shaker located along the structural centerline at the base floor. Four accelerometers (mounted at the center of each floor, including the base) were used to record vertical dynamic responses. The signals were collected using a Dactron Spectrabook data acquisition system, discretized into 8192 points with a sampling interval of 3.125 ms (320 Hz). The acceleration signals were checked for sensor bias, detrended, and subjected to standard band-pass filtering to minimize out-of-band contributions. Additional instrumentation details are available in Figueiredo et al. (2009).

Figure 7.

The schematic of the laboratory 3-story frame structure (Figueiredo et al., 2009) (a) and the corresponding modeled 4-DOF shear-type structural system (b).

In this study, nine experimental scenarios of the frame structure were analyzed (Table 3), encompassing both undamaged and damaged conditions. Scenarios 1 to 3 represent the baseline and two mass-variation cases, where 1.2 kg (about 19% of each floor’s mass) was added to the base and first floors. These mass-change scenarios were set to simulate typical operational and environmental variability and are all treated as undamaged conditions. Scenarios 4 to 9 represent damaged states introduced by reducing the stiffness of one or two columns per story by 87.5%, implemented by replacing the original columns with thinner counterparts (half the cross-sectional thickness in the shaking direction). For all scenarios, acceleration-based CCs were extracted and used to validate the proposed model updating method.

Table 3.

Considered experimental scenarios of the 3-story frame structure in case study 2.

Scenarios	Description
1	Baseline
2	Added mass of 1.2 kg at the base
3	Added mass of 1.2 kg on the 1st floor
4	87.5% stiffness reduction in 1 column of the 1st inter-story
5	87.5% stiffness reduction in 2 columns of the 1st inter-story
6	87.5% stiffness reduction in 1 column of the 2nd inter-story
7	87.5% stiffness reduction in 2 columns of the 2nd inter-story
8	87.5% stiffness reduction in 1 column of the 3rd inter-story
9	87.5% stiffness reduction in 2 columns of the 3rd inter-story

The experimental frame was modeled as a 4-DOF shear-type system, as shown in Figure 7(b), consistent with the structural configuration used in the 8-DOF model of Case Study 1. Classical modal damping was assumed, and only translational modes were considered, as the floor-mounted accelerometers along the centerline are insensitive to torsional motion.

In the model, the 1st DOF represents the base floor where external excitation was applied. The measured base excitation from the experiment was directly used as input. The stiffness $k_{0}$ and damping ratio $ξ_{0}$ between the 1^st DOF and the ground of the modeled 4-DOF system serve to account for the rigid body motion of the frame structure, to represent sliding friction and were tuned to account for energy dissipation at the base.

Parameter identification of frame baseline condition

According to the simulation settings, the 4-DOF model comprises 4 sets of lumped masses $m_{d}$ , modal damping ratios $ξ_{d}$ , and stiffness values $k_{d}$ , for $d = 0, 1, 2, 3$ , where $d = 0$ representing the base floor. Stiffness value $k_{d}^{(s 1)} (d = 1, 2, 3)$ in Scenario 1 (baseline) were initialized from a normal distribution with a mean of 500 kN/m and a 10% standard deviation, reflecting material property variability and uncertainties, based on the aluminum density (2.7 g/cm³) and previous research (Sun and Betti, 2015; Zhang and Sun, 2021). Here, the subscript “(s1)” denotes Scenario 1, representing the baseline condition. The standard deviation accounts for potential inaccuracies in structural material properties and other uncertainties. The base stiffness $k_{0}$ was fixed at 10 N/m to represent sliding friction at the base-rail interface, following the settings established in (Sun and Betti, 2015). The total structural mass under the baseline condition $m_{t o t a l}^{(s 1)}$ , and for Scenarios 4–9, $m_{t o t a l}^{(s 4 - 9)}$ , was about 26 kg. For mass-variation scenarios (Scenarios 2–3), the total mass $m_{t o t a l}^{(s 2 - 3)}$ increased to about 27.2 kg. A global mass constraint was imposed: $\sum_{d = 0}^{3} m_{d} = m_{t o t a l}$ . For Scenario 1 (baseline), each $m_{d}$ was initialized with a mean of 6.5 kg (i.e., 26/4) and a standard deviation of 0.65 kg; each damping ratio $ξ_{d}^{(s 1)}$ was initialized with a value of 3.5%, reflecting typical damping levels in steel/aluminum frame structures (2–5%), as discussed in Methodology.

In this case, although the base stiffness $k_{0}$ and damping ratio $ξ_{0}$ are treated as independent parameters primarily to capture the rigid-body friction effects at the base. Once identified under the baseline condition, these parameters are fixed in subsequent damage-scenario analyses to ensure model consistency and to reduce computational complexity during parameter identification.

The structural parameters under the baseline condition were first identified using Approach 1, which directly estimates parameter values, as in Case Study 1. For all subsequent undamaged and damaged scenarios, Approach 2 was employed to estimate percentage changes relative to the identified baseline parameters. Each scenario in the experimental dataset includes 50 time-history realizations of both acceleration responses and input forces. To ensure reliability, the identification process was repeated 20 times per scenario using randomly selected 20 realizations without replacement. The mean, maximum, and minimum of the identified parameters across these runs were reported, as shown in Table 4.

Table 4.

Identified mass, stiffness, damping ratios of the experimental frame structure system under its baseline condition (s1). The values in brackets indicate the min and max of either the initialized parameters (the 1st row) or the identified results using the 5 different optimizers, obtained from 20 runs of each. The values in bold represent the mean of the identified results across these 20 runs.

	$m_{0}^{(s 1)}$	$m_{1}^{(s 1)}$	$m_{2}^{(s 1)}$	$m_{3}^{(s 1)}$	$k_{1}^{(s 1)}$	$k_{2}^{(s 1)}$	$k_{3}^{(s 1)}$	$ξ_{1}^{(s 1)}$	$ξ_{2}^{(s 1)}$	$ξ_{3}^{(s 1)}$
Initialized parameter ranges or values	[5.22, 7.59]	[5.19, 7.60]	[5.38, 7.42]	[4.98, 7.75]	[393.74, 615.08]	[397.41, 578.94]	[396.86, 609.56]	0.035	0.035	0.035
GA	7.24 [6.49, 8.01]	7.43 [6.78, 8.23]	6.20 [5.57, 6.74]	5.25 [4.54, 6.03]	452.29 [410.76, 495.26]	429.17 [386.52, 467.33]	356.21 [313.42, 398.25]	0.064 [0.056, 0.073]	0.010 [0.006, 0.014]	0.010 [0.005, 0.014]
PSO	7.27 [6.61, 7.84]	7.45 [6.91, 7.93]	6.18 [5.73, 6.76]	5.23 [4.80, 5.78]	450.76 [419.35, 482.34]	427.54 [399.24, 458.18]	357.68 [328.24, 383.62]	0.059 [0.053, 0.068]	0.010 [0.007, 0.013]	0.011 [0.007, 0.014]
CMA-ES	7.31 [7.02, 7.63]	7.49 [7.12, 7.81]	6.16 [5.85, 6.41]	5.21 [4.82, 5.43]	453.11 [435.26, 471.43]	422.06 [406.35, 449.43]	346.21 [324.15, 369.23]	0.063 [0.057, 0.068]	0.009 [0.007, 0.011]	0.009 [0.007, 0.011]
LSHADE	7.33 [7.09, 7.58]	7.48 [7.31, 7.63]	6.12 [5.90, 6.33]	5.10 [4.73, 5.46]	450.03 [424.82, 475.16]	422.33 [402.14, 445.02]	331.45 [310.26, 350.88]	0.062 [0.057, 0.066]	0.010 [0.007, 0.013]	0.010 [0.006, 0.014]
LSHAD-SPACMA	7.32 [7.14, 7.48]	7.48 [7.34, 7.61]	6.11 [5.93, 6.28]	5.08 [4.93, 5.28]	450.14 [438.35, 461.54]	420.17 [410.28, 431.67]	329.18 [316.27, 343.56]	0.061 [0.057, 0.065]	0.010 [0.007, 0.012]	0.010 [0.007, 0.013]

In addition to the LSHADE-SPACMA algorithm integrated within the CC-based FEMU framework, the baseline parameters of the frame were also identified using the same 5 benchmark optimization algorithms considered in Case Study 1, namely the traditional DE, LSHADE, CMA ES, PSO, and GA (Table 4). This consistent benchmarking enables a comprehensive comparison of algorithmic performance on the formulated CC-based objective function. Examining the mean values of the identified stiffness, which is the most critical parameter for structural condition assessment, reveals that all algorithms yield similar results, with the GA exhibiting the largest variance. For damping and mass, the DE-based algorithms (DE, LSHADE, and LSHADE-SPACMA) and CMA-ES yielded consistent results, whereas the PSO and GA exhibited notable deviations.

In the analysis of this experimental frame, the true structural parameters are unknown, preventing direct evaluation of algorithmic accuracy. Therefore, the key criterion is the algorithm’s ability to produce stable and consistent identification results across multiple runs, with achieving properly low variance. As shown in Table 4, the LSHADE-SPACMA and LSHADE demonstrate the highest stability, with the LSHADE-SPACMA showing slightly better robustness, consistent with results from Case Study 1. Accordingly, the baseline parameters identified using LSHADE-SPACMA are adopted as the reference for all subsequent damage scenario analyses, ensuring reliability in tracking structural parameter variations.

Identification of frame parameter changes

Using the baseline parameters identified by the LSHADE-SPACMA algorithm as a reference, Approach 2 was applied to estimate the percentage changes in structural parameters for the remaining scenarios. For Scenarios 2–3 (mass variation) and Scenarios 4–9 (stiffness reduction), all mass, stiffness, and damping parameters were initialized to the mean baseline values. Each scenario was analyzed over 20 runs, each using a different realization of the response data, to ensure robust and reliable identification results. Figure 8 presents the identified percentage changes in mass, stiffness, and damping ratios for Scenarios 2–9.

Figure 8.

Identified changes in (a) mass, (b) stiffness, and (c) damping ratios for Scenarios 2 to 9, expressed as percentages relative to the baseline values identified by LSHADE-SPACMA (Table 4, last row). Bars show the mean values over 20 runs; error bars indicate the minimum and maximum. The black reference line marks the baseline (100%). In (a), the red line denotes the added-mass target (approximately 116%). In (b), the red lines indicate reference stiffness reductions for damage in one column (Scenarios 4, 6, 8) and two columns (Scenarios 5, 7, 9).

As shown in Figure 8(a), the added 1.2 kg mass at the base and first floor is accurately detected, with average identified increases of 115.76% at $m_{0}$ (baseline: 7.32 kg) and 120.02% at $m_{1}$ (baseline: 7.48 kg). These results closely match the theoretical mass increase of approximately 116.4%–116.5%, confirming the method’s capability to reliably detect and quantify mass changes.

As shown in Figure 8(b), the imposed 87.5% stiffness reductions, corresponding to 78.21% and 56.24% of the baseline stiffness for single and double column damage cases, are identified with reasonable accuracy and stability across 20 independent runs in Scenarios 4 – 9. Although the damage locations and relative severities are consistently detected, the identified reductions tend to be slightly underestimated. This discrepancy likely arises from the coupling effects between stiffness and damping during the identification process, as discussed in Case Study 1.

As shown in Figure 8(c), the identified damping ratios vary across floors, with some increasing and others decreasing. In most cases, the total damping is higher than the baseline, and in certain scenarios, all floors exhibit elevated damping values. This suggests a compensatory effect within the optimization process, where increased damping partially offsets the dynamic changes caused by the stiffness reductions and potential energy dissipation of the frame.

As previously noted, elevated damping ratios can reduce damped natural frequencies, mimicking the dynamic effects of stiffness reduction. This facilitates closer alignment between the simulated and measured CC trends in damaged scenarios. While the identified changes, primarily in stiffness and marginally in damping, represent plausible solutions within the optimization process, they may not reflect a globally optimal set, particularly since no actual changes in damping or material properties were introduced. Nonetheless, these slight variations in damping do not materially affect the accuracy of stiffness identification. The developed CC-based FEMU method remains effective in reliably detecting and quantifying local stiffness reductions across inter-story levels.

To further validate the CC-based FEMU method, the consistency between the simulated CCs from the updated 4-DOF FEM model and the measured CCs from the frame structure was evaluated. This comparison, shown in Figure 9 for the 1^st and 3^rd floors across baseline and damaged scenarios, demonstrates strong agreement in both trend and magnitude. As shown, following the optimization using the weighted-error objective function designed to suppress excitation and noise effects (see Methodology), the simulated CCs closely replicate the measured responses. These results confirm that the proposed method effectively captures the CC variations governed by key structural characteristics, i.e., natural frequencies, damping ratios, and mode shapes, which are directly influenced by fundamental parameters, particularly stiffness.

Figure 9.

Comparison between the measured CCs of the experimental 4-DOF frame and the simulated CCs of the corresponding modeled 4-DOF system across the undamaged and damaged scenarios.

This case study demonstrates the effectiveness of the proposed CC-based FEMU method in accurately identifying structural parameters and detecting changes, particularly local stiffness reductions and added mass. By optimizing FEM parameters to align simulated CCs with measured ones, the method reliably captures the dynamic behavior of the frame, offering a robust and efficient strategy for structural parameter identification and condition assessment.

Conclusions

This study presents a novel FEMU method that integrates the CCs derived from structural acceleration responses with a customized LSHADE-SPACMA evolutionary optimization framework. It addresses critical challenges of traditional modal updating, including high sensitivity to noise, reliance on expert intervention, and computational inefficiency, while filling a research gap in utilizing response cepstrum representations for model updating. These advantages make it well-suited for real-world SHM applications. The main conclusions are as follows:

(1) The use of CCs as compact, sequential representations of structural dynamic responses effectively captures essential modal characteristics of natural frequencies, damping ratios, and mode shapes. These CCs are inherently robust to noise and variability in excitations, and enable the construction of a physically motivated objective function based on a location-weighted discrepancies between simulated and measured CCs across recording locations. The formulation establishes a direct link between CC variations and structural parameter changes, ensuring reliable identification and offering a scientifically grounded, scalable alternative to traditional modal updating approaches.

(2) The CCs are extracted using a fully automated digital-signal-processing pipeline applied directly to time-domain acceleration data. This removes the need for intermediate modal analysis or manual feature extraction, enabling an efficient one-step parameter identification process. The method streamlines the workflow while preserving effective exploitation of dynamic characteristics. Consequently, the CC extraction process and the location-weighted objective function can be readily integrated into the FEMU framework, providing a robust and high-performance alternative to conventional objective functions that rely on manually extracted modal characteristics from traditional system identification procedures.

(3) The proposed framework was rigorously validated using both numerical and experimental studies of an 8-DOF simulated structure and a 3-story experimental frame. Results show that the integration of the LSHADE-SPACMA algorithm with customized, problem-specific strategies for initializing and constraining structural parameters of stiffness and damping achieves strong effectiveness and rationality in the proposed CC-based FEMU framework. Comparative evaluations within the framework show that the LSHADE-SPACMA, combined with the proposed strategies, outperforms other metaheuristics including the LSHADE, CMA-ES, PSO, and GA, in the high-dimensional parameter spaces typical of multi-DOF shear-type systems.

(4) Across multiple validation runs on both case study structures, the proposed method accurately identify baseline structural parameters and consistently detected and quantified local stiffness reductions across a range of damage scenarios. The method consistently maintained its accuracy under common stiffness–damping coupling and additive noise, demonstrating strong resilience to real-world disturbances. The method’s stability and adaptability make it particularly effective for SHM applications under uncertain or varying baseline conditions.

To facilitate reproducibility and future research, the MATLAB codes implementing the developed Cepstrum-based LSHADE-SPACMA hybrid FEMU framework have been made openly available on GitHub at https://github.com/ll3097/CCs-LSHADE-SPACMA-FEMU.git.

Discussion

Given the growing complexity of structural environments, future work will focus on improving the robustness of the proposed FEMU method under more variable and challenging excitation conditions, including non-stationary and harmonic inputs. A central goal is to assess whether the method can maintain robust identification despite excitation uncertainties and environmental disturbances. To enhance the resilience of the CC-based objective, advanced feature enhancement techniques including autoencoders and nonlinear manifold learning will be explored to isolate the most informative structures in the CCs. These developments aim to enhance the scalability, speed, and intelligence of the method in complex, data-rich SHM scenarios.

Footnotes

ORCID iDs

Lechen Li

Hongwei Zhang

Wenbin Ye

Funding

This work was primarily supported by the National Key Research and Development Program of China (Grant No. 2024YFC3214901). Additional support was provided by the Fundamental Research Funds for the Central Universities (Grant No. B240201069).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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